Polytope of Type {12,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*432b
if this polytope has a name.
Group : SmallGroup(432,301)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 36, 108, 18
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {12,6,2} of size 864
   {12,6,3} of size 1296
   {12,6,4} of size 1728
   {12,6,4} of size 1728
Vertex Figure Of :
   {2,12,6} of size 864
   {4,12,6} of size 1728
   {4,12,6} of size 1728
   {4,12,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,6}*216b
   3-fold quotients : {12,6}*144a
   4-fold quotients : {6,6}*108
   6-fold quotients : {6,6}*72a
   9-fold quotients : {12,2}*48, {4,6}*48a
   18-fold quotients : {2,6}*24, {6,2}*24
   27-fold quotients : {4,2}*16
   36-fold quotients : {2,3}*12, {3,2}*12
   54-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {24,6}*864b, {12,12}*864c
   3-fold covers : {12,18}*1296a, {36,6}*1296b, {12,6}*1296a, {12,6}*1296b, {12,18}*1296b, {36,6}*1296f, {12,18}*1296c, {36,6}*1296g, {12,6}*1296g
   4-fold covers : {48,6}*1728b, {12,12}*1728c, {12,24}*1728d, {24,12}*1728d, {12,24}*1728f, {24,12}*1728f, {12,12}*1728j, {12,6}*1728b
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 3.
      6 facets:
         6 of {12}*24
      20 vertex figures:
         8 of {6}*12
         12 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1> of order 3.
      10 facets:
         6 of {4}*8
         4 of {12}*24
      12 vertex figures:
         12 of {6}*12

Permutation Representation (GAP) : 
s0 := (  2,  3)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)( 37, 46)( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)( 55, 82)( 56, 84)( 57, 83)( 58, 85)( 59, 87)( 60, 86)( 61, 88)( 62, 90)( 63, 89)( 64,100)( 65,102)( 66,101)( 67,103)( 68,105)( 69,104)( 70,106)( 71,108)( 72,107)( 73, 91)( 74, 93)( 75, 92)( 76, 94)( 77, 96)( 78, 95)( 79, 97)( 80, 99)( 81, 98);;
s1 := (  1, 64)(  2, 65)(  3, 66)(  4, 72)(  5, 70)(  6, 71)(  7, 68)(  8, 69)(  9, 67)( 10, 55)( 11, 56)( 12, 57)( 13, 63)( 14, 61)( 15, 62)( 16, 59)( 17, 60)( 18, 58)( 19, 73)( 20, 74)( 21, 75)( 22, 81)( 23, 79)( 24, 80)( 25, 77)( 26, 78)( 27, 76)( 28, 91)( 29, 92)( 30, 93)( 31, 99)( 32, 97)( 33, 98)( 34, 95)( 35, 96)( 36, 94)( 37, 82)( 38, 83)( 39, 84)( 40, 90)( 41, 88)( 42, 89)( 43, 86)( 44, 87)( 45, 85)( 46,100)( 47,101)( 48,102)( 49,108)( 50,106)( 51,107)( 52,104)( 53,105)( 54,103);;
s2 := (  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 13)( 11, 15)( 12, 14)( 17, 18)( 19, 22)( 20, 24)( 21, 23)( 26, 27)( 28, 31)( 29, 33)( 30, 32)( 35, 36)( 37, 40)( 38, 42)( 39, 41)( 44, 45)( 46, 49)( 47, 51)( 48, 50)( 53, 54)( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 67)( 65, 69)( 66, 68)( 71, 72)( 73, 76)( 74, 78)( 75, 77)( 80, 81)( 82, 85)( 83, 87)( 84, 86)( 89, 90)( 91, 94)( 92, 96)( 93, 95)( 98, 99)(100,103)(101,105)(102,104)(107,108);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) : 
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(108)!(  2,  3)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)( 37, 46)( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)( 55, 82)( 56, 84)( 57, 83)( 58, 85)( 59, 87)( 60, 86)( 61, 88)( 62, 90)( 63, 89)( 64,100)( 65,102)( 66,101)( 67,103)( 68,105)( 69,104)( 70,106)( 71,108)( 72,107)( 73, 91)( 74, 93)( 75, 92)( 76, 94)( 77, 96)( 78, 95)( 79, 97)( 80, 99)( 81, 98);
s1 := Sym(108)!(  1, 64)(  2, 65)(  3, 66)(  4, 72)(  5, 70)(  6, 71)(  7, 68)(  8, 69)(  9, 67)( 10, 55)( 11, 56)( 12, 57)( 13, 63)( 14, 61)( 15, 62)( 16, 59)( 17, 60)( 18, 58)( 19, 73)( 20, 74)( 21, 75)( 22, 81)( 23, 79)( 24, 80)( 25, 77)( 26, 78)( 27, 76)( 28, 91)( 29, 92)( 30, 93)( 31, 99)( 32, 97)( 33, 98)( 34, 95)( 35, 96)( 36, 94)( 37, 82)( 38, 83)( 39, 84)( 40, 90)( 41, 88)( 42, 89)( 43, 86)( 44, 87)( 45, 85)( 46,100)( 47,101)( 48,102)( 49,108)( 50,106)( 51,107)( 52,104)( 53,105)( 54,103);
s2 := Sym(108)!(  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 13)( 11, 15)( 12, 14)( 17, 18)( 19, 22)( 20, 24)( 21, 23)( 26, 27)( 28, 31)( 29, 33)( 30, 32)( 35, 36)( 37, 40)( 38, 42)( 39, 41)( 44, 45)( 46, 49)( 47, 51)( 48, 50)( 53, 54)( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 67)( 65, 69)( 66, 68)( 71, 72)( 73, 76)( 74, 78)( 75, 77)( 80, 81)( 82, 85)( 83, 87)( 84, 86)( 89, 90)( 91, 94)( 92, 96)( 93, 95)( 98, 99)(100,103)(101,105)(102,104)(107,108);
poly := sub<Sym(108)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) : 
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
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